Spin, being a purely quantum mechanical effect, must be introduced in the kinetic theory via quantum field theory. Hence, our starting point is the Wigner function for spin-12 particles and its Clifford-algebra decomposition [1]}, [2]}, [3]}, [4]}, [5]}, [6]},
\(\mathcal {W}(x,k) &= \frac{1}{4} \left[ \mathcal {F}(x,k) \!+\! i \gamma _5 \mathcal {P}(x,k) \!+\! \gamma ^\mu {\cal V}_{\mu }(x,k) \!+\! \gamma _5 \gamma ^\mu {\cal A}_{\mu }(x,k) \!+\! \Sigma ^{\mu \nu } {\cal S}_{\mu \nu }(x,k) \right]\!, \) 
